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Full text of "Mathematical And Physical Papers - Iii"

GIST THE  CONDUCTION OF HEAT IN  CRYSTALS.            205
1. Let P be any point of a solid body, homogeneous or heterogeneous, crystallized or imcrystallized; suppose the temperature of the body to vary from point to point, and let dS be an elementary plane area drawn through P in a given direction. The quantity of heat which passes across the element dS in the elementary time dt will be ultimately proportional to dSdt, and may be expressed by fdSdt. This quantity / is the flux of heat referred to a unit of surface. Its value will depend upon the time, upon the position of the point P, and upon the direction of the elementary plane drawn through P. For the present, suppose the time and the position of the point P given, and consider only the variation of / in different directions about P.
If we suppose the values of/ given in the direction of each of .three planes, rectangular or not, passing through P, its value in the direction of any fourth plane follows. For make P the vertex of a triangular pyramid, of which the sides are in the direction of the first three planes, and the base is parallel to the fourth, and then conceive the base, remaining parallel to itself, to approach indefinitely to P. The quantity of heat gained by the pyramid during the time dt is equal to the quantity which enters by the faces, diminished by the quantity which escapes by the base. Now when the pyramid is indefinitely diminished, the gain of heat in a given indefinitely short time will vary ultimately as the volume, or as the cubes of homologous lines, whereas the quantity which passes across any one of the four faces of the pyramid will vary ultimately as the area of the face, and therefore as the squares of homologous lines. Hence in the limit the quantity of heat which escapes by the base will be equal to the sum of the quantities which enter by the sides, and consequently if the flux across each be given, the flux across the base is determinate.
In particular, if we suppose the medium referred to the rectangular axes of x, y, z, and if fx, fy, fz be the fluxes across three planes drawn through P in directions perpendicular to the axes of x, y, z; f the flux across a plane drawn in any other direction through P; I, m} n the cosines of the angles which the normal to this plane makes with the axes, we have
(1).